Complete The Synthetic Division To Find The Quotient Of $3x^3 - 25x^2 + 12x - 32$ And $x - 8$.Drag The Numbers To The Correct Locations On The Image. Not All Numbers Will Be Used.Numbers: 4, 32, 0, -8, 8,

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Introduction

Synthetic division is a shortcut method used to divide polynomials by linear factors. It is a simplified version of the long division method and is particularly useful when dividing polynomials by a linear factor of the form (x - a). In this article, we will use synthetic division to find the quotient of the polynomial $3x^3 - 25x^2 + 12x - 32$ and the linear factor $x - 8$.

The Synthetic Division Process

The synthetic division process involves the following steps:

  1. Write down the coefficients of the polynomial in a row, with the coefficient of the highest degree term first.
  2. Write down the value of the linear factor in the form (x - a) below the row of coefficients.
  3. Bring down the first coefficient.
  4. Multiply the value of the linear factor by the first coefficient and write the result below the second coefficient.
  5. Add the second coefficient and the result from the previous step.
  6. Repeat steps 4 and 5 until all coefficients have been used.
  7. The final result is the quotient of the polynomial and the linear factor.

Example: Synthetic Division of $3x^3 - 25x^2 + 12x - 32$ and $x - 8$

To find the quotient of the polynomial $3x^3 - 25x^2 + 12x - 32$ and the linear factor $x - 8$, we will use the synthetic division process.

Step 1: Write Down the Coefficients and the Value of the Linear Factor

The coefficients of the polynomial are 3, -25, 12, and -32. The value of the linear factor is 8.

Step 2: Bring Down the First Coefficient

The first coefficient is 3. We will bring it down to the next row.

Step 3: Multiply the Value of the Linear Factor by the First Coefficient

The value of the linear factor is 8, and the first coefficient is 3. We will multiply them together to get 24.

Step 4: Add the Second Coefficient and the Result from the Previous Step

The second coefficient is -25. We will add it to the result from the previous step, which is 24. The result is -1.

Step 5: Repeat Steps 3 and 4 for the Remaining Coefficients

We will repeat steps 3 and 4 for the remaining coefficients.

3 -25 12 -32
8 3 -1 4 32
24 -24 40 -256
-1 4 32

Step 6: Write Down the Quotient

The final result is the quotient of the polynomial and the linear factor. The quotient is 3x^2 - x + 4.

Conclusion

In this article, we used synthetic division to find the quotient of the polynomial $3x^3 - 25x^2 + 12x - 32$ and the linear factor $x - 8$. The quotient is 3x^2 - x + 4. Synthetic division is a useful shortcut method for dividing polynomials by linear factors and can be used to find the quotient of two polynomials.

Discussion

Synthetic division is a powerful tool for dividing polynomials by linear factors. It is particularly useful when dividing polynomials by a linear factor of the form (x - a). The process involves bringing down the first coefficient, multiplying the value of the linear factor by the first coefficient, and adding the second coefficient and the result from the previous step. We repeated this process for the remaining coefficients to find the quotient of the polynomial and the linear factor.

Key Takeaways

  • Synthetic division is a shortcut method for dividing polynomials by linear factors.
  • The process involves bringing down the first coefficient, multiplying the value of the linear factor by the first coefficient, and adding the second coefficient and the result from the previous step.
  • We used synthetic division to find the quotient of the polynomial $3x^3 - 25x^2 + 12x - 32$ and the linear factor $x - 8$.
  • The quotient is 3x^2 - x + 4.

Frequently Asked Questions

  • What is synthetic division? Synthetic division is a shortcut method for dividing polynomials by linear factors.
  • How do I use synthetic division to find the quotient of two polynomials? To use synthetic division, you will need to bring down the first coefficient, multiply the value of the linear factor by the first coefficient, and add the second coefficient and the result from the previous step. You will repeat this process for the remaining coefficients to find the quotient of the polynomial and the linear factor.
  • What is the quotient of the polynomial $3x^3 - 25x^2 + 12x - 32$ and the linear factor $x - 8$? The quotient is 3x^2 - x + 4.

References

Image Credits

  • The image used in this article is a public domain image from Wikipedia.

License

This article is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

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Frequently Asked Questions

Q: What is synthetic division?

A: Synthetic division is a shortcut method for dividing polynomials by linear factors. It is a simplified version of the long division method and is particularly useful when dividing polynomials by a linear factor of the form (x - a).

Q: How do I use synthetic division to find the quotient of two polynomials?

A: To use synthetic division, you will need to bring down the first coefficient, multiply the value of the linear factor by the first coefficient, and add the second coefficient and the result from the previous step. You will repeat this process for the remaining coefficients to find the quotient of the polynomial and the linear factor.

Q: What is the difference between synthetic division and long division?

A: Synthetic division is a shortcut method for dividing polynomials by linear factors, while long division is a more general method for dividing polynomials by any polynomial of the form (x - a). Synthetic division is particularly useful when dividing polynomials by a linear factor of the form (x - a).

Q: Can I use synthetic division to divide polynomials by any polynomial?

A: No, synthetic division is only used to divide polynomials by linear factors of the form (x - a). If you need to divide a polynomial by a polynomial of a higher degree, you will need to use the long division method.

Q: How do I know if a polynomial can be divided by a linear factor?

A: A polynomial can be divided by a linear factor if the linear factor is a factor of the polynomial. You can use the remainder theorem to check if a polynomial is divisible by a linear factor.

Q: What is the remainder theorem?

A: The remainder theorem states that if a polynomial f(x) is divided by a linear factor (x - a), then the remainder is equal to f(a).

Q: How do I use the remainder theorem to check if a polynomial is divisible by a linear factor?

A: To use the remainder theorem, you will need to evaluate the polynomial at the value of the linear factor. If the result is zero, then the polynomial is divisible by the linear factor.

Q: What is the quotient of the polynomial $3x^3 - 25x^2 + 12x - 32$ and the linear factor $x - 8$?

A: The quotient is 3x^2 - x + 4.

Q: Can I use synthetic division to find the remainder of a polynomial division?

A: Yes, synthetic division can be used to find the remainder of a polynomial division. The remainder is the value of the polynomial at the value of the linear factor.

Q: How do I use synthetic division to find the remainder of a polynomial division?

A: To use synthetic division to find the remainder, you will need to bring down the first coefficient, multiply the value of the linear factor by the first coefficient, and add the second coefficient and the result from the previous step. You will repeat this process for the remaining coefficients to find the remainder.

Q: What is the remainder of the polynomial $3x^3 - 25x^2 + 12x - 32$ and the linear factor $x - 8$?

A: The remainder is 32.

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License

This article is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.